T. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for stationary signals, is reported in [70] To alleviate the cost of computing the SVD, alternative but computationally less demanding decompositions of the form X = UExV , where Ex is not diagonal any more, are gaining interest. Recent developments are the rank revealing QR factorization [135] which can be updated [136] and the rank revealing URV decomposition [137] where E = R is upper triangular. In this 72 decomposition, R has a block decomposition into four blocks, such that and R22 both have small Frobenius norms, and the smallest singular value of R is of the order of the ....

T. Chan, "Rank revealing qr factorizations," Lin. Alg. Appl., vol. 88/89, pp. 67-82, 1987.

....been investigated. For QR factorization we try to choose a permutation matrix Pi so that A Pi = QR is a rank revealing QR factorization. The standard column pivoting strategy [69, x5.4.1] tends to be rank revealing, but it can completely fail to reveal near rank deficiency. Foster [63] and Chan [25] develop iterative algorithms for computing rankrevealing QR factorizations; for both algorithms the bounds that show by how much, in the worst case, the factorizations may fail to reveal the rank contain factors exponential in the rank deficiency. Both methods need a condition estimator, to ....

Tony F. Chan. Rank revealing QR factorizations. Linear Algebra and Appl., 88/89:67--82, 1987.

....scalars c, sgn(cM) sgn(c)sgn(M ) 5. sgn(TMT ) Tsgn(M)T for any nonsingular T . By property 2, an orthogonal basis for the left invariant subspace of M which has a dimension r is given by the first r columns of the orthogonal matrix in a rank revealing QR decomposition of (Z Gamma I) [11]. Alternative methods are presented in [3] and [4] The above definition for the matrix sign does not lend itself to an efficient computation but there are several ways of evaluating the matrix sign function. The simplest iteration scheme is Newton s method applied to sgn(M) I: Z 0 = M; Z ....

T. F. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.

..... Gamma1 1 3 7 7 7 7 7 5 ; 13) which has a singular value of the order of magnitude of 2 Gamman . Therefore great care has to be put in finding the nullity of the triangular matrix R 1 . This can e.g. be done by using the rank revealing QR decomposition made popular by T. Chan [4]. This technique allows one to update the R 1 factor of the QR decomposition using left and right unitary transformations in order to display the nullity (say d 1 ) of R as follows : U 1 Delta R 1 Delta V 1 : R = 2 6 6 6 6 6 6 6 6 4 0 d 1 Delta Delta Delta r d 1 1;d 1 1 ....

....O(n 3 ) The initial QR factorization (11) is O(n 3 ) and is done only once. In each subsequent step one then performs the updating transformations (14) for obtaining a rank revealing triangular factor R. Assuming the nullity is d 1 at step 1, this involves only O(d 1 n 2 ) operations (see [4]) Then one needs to update the pencil Q Gamma R to yield Q Gamma R as given in (20) It is shown in [3] that the updating transformation U 2 can be performed with d 1 (n Gamma d 1 1 2 ) Givens transformations and V 2 with d 1 (n Gamma d 1 ) Givens transformations. Indeed, the ....

T. Chan, Rank revealing QR-factorizations, Linear Algebra & Applications 88/89 (1987) 67-82.

....presence of the terms d P k=0 a k r s;N Gammak in (2.1.10) the rank of Phi may exceed r when OE is not proper. However, Phi s = V 0 s Gamma V s when s r 1 (see 2.1. 14) Since the pole rank of OE may differ from the rank of Phi; numerical methods for finding rank, such as those of [C], are not directly relevant. We search for the pole rank by examining relative singular values (the reciprocals of the spectral condition numbers [GvL: 2.7.2] W: 2.30] s = s ( Phi) oe s ( Phi s ) oe 1 ( Phi s ) where oe j ( Phi s ) is the j th singular value of Phi s ; for r 1 ....

Chan, T. 1987. Rank Revealing QR factorizations. Linear Algebra Appl. 88/89: 67-82.

....presence of the terms d P k=0 a k r s;N Gammak in (2.1.10) the rank of Phi may exceed r when OE is not proper. However, Phi s = V 0 s Gamma V s when s r 1 (see 2.1. 14) Since the pole rank of OE may differ from the rank of Phi; numerical methods for finding rank, such as those of [C], are not directly relevant. We search for the pole rank by examining relative singular values (the reciprocals of the spectral condition numbers [GvL: 2.7.2] W: 2.30] s = s ( Phi) oe s ( Phi s ) oe 1 ( Phi s ) where oe j ( Phi s ) is the j th singular value of Phi s ; for r 1 ....

Chan, T. 1987. Rank Revealing QR factorizations. Linear Algebra Appl. 88/89: 67-82.

....rank k, then the initial triangular factorization requires O(mn 2 ) ops, while the rank revealing step only requires O( n k)n 2 ) ops if k n, and O(kn 2 ) ops if k n. The updating can always be done in O(n 2 ) ops, when implemented properly. We refer to the original papers [9], 10] 16] 18] 19] 23] 31] 32] for details about the algorithms. For structured matrices (e.g. Hankel and Toeplitz matrices) the initial triangular factorization in the RRQR and UTV algorithms has the same complexity as the rank revealing step, namely, O(mn) ops; see [7, x8.4.2] for ....

T. F. Chan, Rank-revealing QR factorizations, Lin. Alg. Appl., 88/89 (1987), pp. 67-82.

....performance on parallel computers is normally more efficient when the orthogonal factorization is carried out without pivoting for numerical stability. Column interchanges are also (explicitly or implicitly) needed if one wants to compute a rank revealing QR factorization; see, for example, Chan [5], Foster [10] or Pierce and Lewis [25] Developing a parallel rank revealing code for sparse matrices will be one of the topics for future work. Both column and row interchanges must be used in order to preserve better the sparsity of matrix A during the orthogonal decomposition. Finding a good ....

T. F. Chan, Rank revealing QR factorizations, Lin. Alg. Appl., Vol. 88/89 1987, pp. 67-82.

....In the past few years, a number of other methods have been developed to alleviate the computational burden of the SVD, yet retaining important information such as rank and principal subspaces. Some of these techniques are the URV decomposition [2 ] and the rank revealing QR decomposition (RRQR) [3 , 4, 5]. Recently, there has been an increased interest in updating techniques for the SVD and URV decomposition, which converge to the exact SVD or URV under certain stationarity conditions [6 , 7] It should be noted that all these decompositions require O(am 2 n) operations, for an m n matrix, ....

T.F. Chan, "Rank revealing QR factorizations," Lin. Alg. Appl., vol. 88/89, pp. 67--82, 1987. 12

....Linear Algebra, pp. 274 278, Snowbird, Utah, June 1994. 1 2 A.J. van der Veen Other methods to alleviate the computational burden of the SVD while retaining important information such as rank and principal subspaces are the URV decomposition [5] and the rank revealing QR decomposition (RRQR) [2, 1]. The Schur method requires approximately the same number of operations, but has a simpler and more uniform dependence structure. No condition estimation or other global operations are needed, and the number of operations to determine the column space of the approximant is independent of the ....

T.F. Chan, Rank revealing QR factorizations, Lin. Alg. Appl., 88/89 (1987), pp. 67--82.

....invariant subspace is immediate by property 2. A numerically stable algorithm of finding this subspace out of the matrix sign function is given below based on [7] Let us assume r eigenvalues of M lie in the open left half plane, counting multiplicities. Now let the rank revealing QR decomposition [13] of the matrix Z l = sgn(M) Gamma I be Z l = Q R Pi, where R is upper triangular, Q is orthogonal, and Pi is a permutation matrix chosen so that the leading r columns of Q span the range space of Z l . By this decomposition, one finds an orthogonal basis for the left invariant subspace of M . ....

T. F. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.

....subspaces is not difficult. Based on [7] we outline below a numerically stable and efficient method for this purpose. Let us assume r eigenvalues of an n Thetan real matrix M lie in the open left half plane counting multiplicities, and let Z = sgn(M ) Also let the rank revealing QR decomposition [14] of the matrix Z Gamma I be Z l 4 = Z Gamma I = Q l R l Pi l ; 55) where R l is upper triangular, Q l is orthogonal, and Pi l is a permutation matrix. Suppose that the permutation Pi l is chosen so that the rank deficiency of Z l is exhibited in R l by a smaller lower right block in norm ....

T. F. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.

.... I) is equal to the left (right) invariant subspace of M [32] Then find S = sgn(W e ) 49) through the matrix sign function iterations (47) Recall that there are m u eigenvalues of W e in the right half plane and m s eigenvalues in the left half plane. Let the rank revealing QR decomposition [8] of S I be S I = Q r R r Pi r ; 50) where R r is upper triangular, Q r is orthogonal, and Pi r is a permutation matrix. Suppose that Pi r is chosen so that the rank deficiency of S I is exhibited in R r by a smaller lower right block in norm of size m s Theta m s . Then, let V 1 4 = ....

T. F. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.

....= sgn(c)sgn(M ) 5. sgn(TMT Gamma1 ) Tsgn(M)T Gamma1 for any nonsingular T . By property 2, an orthogonal basis for the left invariant subspace of M which has a dimension r is given by the first r columns of the orthogonal matrix in a rank revealing QR decomposition of (Z Gamma I) [11]. Alternative methods are presented in [3] and [4] The above definition for the matrix sign does not lend itself to an efficient computation but there are several ways of evaluating the matrix sign function. The simplest iteration scheme is Newton s method applied to sgn(M) 2 = I: Z 0 = M; Z ....

T. F. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.

....of B is unique. The result of Theorem 3.5.4 holds for the decomposition (3.6.44) with J replaced by B . This proves the existence of a column permutation of B such that the bottom element of R in B s QR decomposition is tiny when B is nearly singular. This result was proved by Chan in [18] and earlier by Golub, Klema and Stewart in [66] Thus twisted factorizations can be rank revealing. Rank revealing LU and QR factorizations have been extensively studied and several algorithms to compute such factorizations exist. Twisted factorizations seem to have been overlooked and may offer ....

....Rank revealing LU and QR factorizations have been extensively studied and several algorithms to compute such factorizations exist. Twisted factorizations seem to have been overlooked and may offer computational advantages. We consider our results outlined above to be stronger than those of Chan [19, 18] and Golub et al. 66] since the permutations we consider are restricted. In particular, as seen in the tridiagonal and Hessenberg case, twisted factorizations respect the sparsity structure of the given matrix, and thus may offer computational advantages in terms of speed and accuracy. We believe ....

T. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.

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T. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.

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T. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987.

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T. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987. 43

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T. Chan. Rank-revealing QR factorizations. Lin. Alg. Appl., 88/89:67{ 82, 1987.

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T. Chan. Rank revealing QR factorizations. Lin. Alg. Appl., 88/89:67--82, 1987. 47

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T. Chan, Rank revealing QR factorizations, Linear Algebra Appl., 88/89 (1987), pp. 67--82.

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T.F. CHAN, Rank revealing QR factorizations. Lin. Alg. Appl. 88/89 (1987), pp. 67-82.

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T. Chan. Rank revealing QR factorizations. Lin. Alg. & Appl., 88/89:67--82, 1987.

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T. Chan. Rank revealing QR factorizations. Lin. Alg. & Appl., 88/89 (1987) 67--82.

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T.F. Chan. Rank revealing QR factorizations. Linear Algebra and its Appl. 88/89 (1987), 67-82.

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